A second but equally important consideration is that the
eigenstates of the cavity (the polariton states) are embedded
in a continuum of photon states
exterior to the cavity which “dress”
the cavity states.[1]}
Within this picture, sketched in Fig. REF , each polariton state, \(\phi \) ,
is
embedded into the continuum by writing it as

|= c|+ ncn|n.


These are formally eigenstates
states of a Schrödinger equation
\((H_{cav} + H_{ext} + \hat{W})|\psi _\mu \rangle = E_\mu |\psi _\mu \rangle \) 
where \(H_{cav}\)  is the cavity Hamiltonian
with eigenstates \(|\phi \rangle \) 
with energies \(E_\phi \) .
\(H_{ext}\)  is the Hamiltonian
for a quasi-continuum of external states
\(|n\rangle \)  with energy
\(E_n = n \delta \) , respectively. Here, integer \(n\in (-\infty ,\infty )\)  indexes the quasi-continuum states
with uniform spacing \(\delta \) . Our final results
will require \(\delta \rightarrow 0\) .
